\subsection*{Substitution ciphers}

\begin{exercise}
\label{ex:classical_crypto:all_keys_affine_ciphers}
Determine the number of possible keys for the affine substitution ciphers.
Is this sufficient to have a secure cryptosystem?
\end{exercise}

\subsection*{Transposition ciphers}

\begin{exercise}
\label{ex:classical_crypto:order_of_group}
Show that for every $\pi$ in $S_n$, there exists an positive integer $m$,
such that $\pi^m$ is the identity map, and such that $m$ divides $n!$.
The smallest such $m$ is called the order of $\pi$.
\end{exercise}

\begin{exercise}
\label{ex:classical_crypto:all_transpositions_Sn}
How many transpositions exist in $S_n$?  Describe the elements of order $2$
in $S_n$ and determine their number.
\end{exercise}

\begin{exercise}
\label{ex:classical_crypto:express_elements_of_Sn_as_compositions}
Show that every element of $S_n$ can be expressed as the composition of at
most $n$ transpositions.
\end{exercise}

\begin{exercise}
\label{ex:classical_crypto:order_of_permutation}
What is the order of a permutation with cycle lengths $d_1,\dots,d_t$?  How
does this solve the previous exercise concerning the order of a permutation?
\end{exercise}

\begin{exercise}
\label{ex:classical_crypto:block_length_columnar_transposition}
What is the block length $m$ of an $(r,s)$-simple columnar transposition?
Describe the permutation.  Hint: it may be easier to describe the permutation
if the index set is $\{0,\dots,m-1\}$.
\end{exercise}

\begin{exercise}
\label{ex:classical_crypto:columnar_transposition_order_2}
Show that the $(r,r)$-simple columnar transposition has order $2$.  What is
the order of the cipher for $(r,s) = (3,5)$?  Determine the permutation in
cycle notation for this cipher.  Determine the permutation in cycle notation
for the $(7,36)$-simple columnar transposition used in this chapter.
\end{exercise}
